Baker–Campbell–Hausdorff formula

The Baker–Campbell–Hausdorff formula: existence

The Baker–Campbell–Hausdorff formula implies that if X and Y are in some Lie algebra g , defined over any field of characteristic 0 like ℝ or ℂ, then

can formally be written as an infinite sum of elements of g . [This infinite series may or may not converge, so it need not define an actual element Z in g .] For many applications, the mere assurance of the existance of this formal expression is sufficent, and an explicit expression for this infinite sum is not needed. This is for instance the case in the Lorentzian construction of a Lie group representation from a Lie algebra representation. Existence can be seen as follows.

We consider the ring

of all non-commuting formal power series with real coefficients in the non-commuting variables X and Y. There is a ring homomorphism from S to the tensor product of S with S over R,

called the coproduct, such that

(The definition of Δ is extended to the other elements of S by requiring R-linearity, multiplicativity and infinite additivity.)

One can then verify the following properties:

The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows: The elements X and Y are primitive, so exp(X) and exp(Y) are grouplike; so their product exp(X)exp(Y) is also grouplike; so its logarithm log(exp(X)exp(Y)) is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated by X and Y.

The universal enveloping algebra of the free Lie algebra generated by X and Y is isomorphic to the algebra of all non-commuting polynomials in X and Y. In common with all universal enveloping algebras, it has a natural structure of a Hopf algebra, with a coproduct Δ. The ring S used above is just a completion of this Hopf algebra.

An alternative, remarkably direct and concise, recursive proof that all homogeneous polynomials in Z are in the Lie algebra is due to Eichler.

An explicit Baker–Campbell–Hausdorff formula

Specifically, let G be a Lie group with Lie algebra g . Let

be the exponential map. The following general combinatorial formula was introduced by Eugene Dynkin (1947),

where the sum is performed over all nonegative values of s i and r i , and the following notation has been used:

Since [A, A] = 0, the term is zero if s n > 1 or if s n = 0 and r n > 1 .

The first few terms are well-known, with all higher-order terms involving [X,Y] and commutator nestings thereof (thus in the Lie algebra):

The above lists all summands of order 5 or lower (i.e. those containing 5 or fewer X's and Y's). Note the X ↔ Y (anti-)/symmetry in alternating orders of the expansion, since Z(Y, X) = −Z(−X,−Y). A complete elementary proof of this formula can be found here.

Selected tractable cases

There is no expression in closed form for an arbitrary Lie algebra, though there are exceptional tractable cases, as well as efficient algorithms for working out the expansion in applications.

For example, if [X,Y] vanishes, then the above formula reduces to X+Y. If the commutator [X,Y] is a scalar (central, cf. the nilpotent Heisenberg group), then all but the first three terms on the right-hand side of the above vanish. This special case may be established directly. This is the degenerate case utilized routinely in quantum mechanics, as illustrated below.

Other forms of the Baker–Campbell–Hausdorff formula, emphasizing expansion in terms of the element Y (and utilizing the linear adjoint endomorphism notation, ad_X Y ≡ [X,Y]), might serve well:

as is evident from the integral formula below. (The coefficients of the nested commutators linear in Y are normalized Bernoulli numbers, outlined below.)

Thus, when the commutator happens to be [X,Y]=sY, for some non-zero s, this formula reduces to just Z = X + sY / (1 − exp(−s)), which then leads to braiding identities such as

or adjoint dilation,

There are numerous such well-known expressions applied routinely in physics. A popular integral formula is

involving the generating function for the Bernoulli numbers,

utilized by Poincaré and Hausdorff.

Matrix Lie group illustration

For a matrix Lie group G ⊂ GL ( n , R ) the Lie algebra is the tangent space of the identity I, and the commutator is simply [X, Y] = XY − YX; the exponential map is the standard exponential map of matrices,

When one solves for Z in

using the series expansions for exp and log one obtains a simpler formula:

The first, second, third, and fourth order terms are:

The main significance of the Baker–Campbell–Hausdorff formula is the non-obvious fact that each of the terms of degree two and higher can be expressed as a linear combination of iterated brackets of X and Y, with rational coefficients, as proven compactly in the above-mentioned Eichler note. As an immediate consequence one obtains the following result about the trace:

Questions of convergence

There exist real 2-by-2 matrices X and Y such that there is no real 2-by-2 matrix Z with

An example is

The formulas above that promise to give expressions for Z in terms of iterated Lie-brackets of X and Y therefore describe formal power series whose convergence is not guaranteed; they are guaranteed to be globally valid only if working in the ring of formal power series in the non-commuting variables X and Y, with the commutator as Lie-bracket.

However, one can show that if we work in a Lie algebra over the field R or C and the elements X and Y are close enough to 0, then the Dynkin formula given above converges absolutely to a Lie algebra element Z with

Concretely, if working with a matrix Lie algebra and ∥ . ∥ is a given submultiplicative matrix norm, then this will happen if

The Zassenhaus formula

A related combinatoric expansion that is useful in dual applications is

where the exponents of higher order in t are likewise nested commutators, i.e., homogeneous Lie polynomials. These exponents, C_n in exp(–tX) exp(t(X+Y)) = Π_n exp(tⁿ C_n), follow recursively by application of the above BCH expansion.

As a corollary of this, the Suzuki–Trotter decomposition follows directly.

An important lemma

Let G be a matrix Lie group and g its corresponding Lie algebra. Let ad_X be the linear operator on g defined by ad_X Y = [X,Y] = XY − YX for some fixed X ∈ g. (The adjoint endomorphism encountered above.) Denote with Ad_A for fixed A ∈ G the linear transformation of g given by Ad_AY = AYA⁻¹.

A standard combinatorial lemma which is utilized in producing the above explicit expansions is given by

so, explicitly,

This formula can be proved by evaluation of the derivative with respect to s of f (s)Y ≡ e^sX Y e^−sX, solution of the resulting differential equation and evaluation at s = 1,

or

For [X,Y] central, i.e., commuting with both X and Y,

Consequently, for g(s) ≡ e^sX e^sY, it follows that

whose solution is

hence the degenerate form already covered above,

More generally, for non-central [X,Y] , the following braiding identity further follows readily,

Application in quantum mechanics

A degenerate form of the Baker–Campbell–Hausdorff formula is useful in quantum mechanics and especially quantum optics, where X and Y are Hilbert space operators, generating the Heisenberg group.

A typical example is the annihilation and creation operators, â and â^†. Their commutator [â^†,â]= −I is central, that is, it commutes with both â and â^†. As indicated above, the expansion then collapses to the semi-trivial degenerate form:

where v is just a complex number.

This example illustrates the resolution of the displacement operator, exp(vâ^†−v^*â), into exponentials of annihilation and creation operators and scalars.

This degenerate Baker–Campbell–Hausdorff formula then displays the product of two displacement operators as another displacement operator (up to a phase factor), with the resultant displacement equal to the sum of the two displacements,

since the Heisenberg group they provide a representation of is nilpotent. The degenerate Baker–Campbell–Hausdorff formula is frequently used in quantum field theory as well.